The Arithmetic of Infinitesimals: John Wallis 1656

Jackie Stedall is amazing. After giving us a history of English algebra up to 1685 and an annotated translation of Thomas Harriot's manuscripts on algebra, here is her translation of John Wallis's famous Arithmetic of Infinitesimals (Arithmetica Infinitorum, first published in 1656). Thank you, Jackie; please never stop.

Wallis's subtitle gives a good summary of what the book is about: "A New Method of Inquiring into the Quadrature of Curves, and other more difficult mathematical problems". Wallis deploys infinitesimal arguments to solve mathematical problems, in particular problems of "quadrature" (which we would describe as "integration", or perhaps as "computing areas"). Though the book is presented in the classical way, as a sequence of Propositions, it is more exploratory than Euclidean in spirit. Wallis is not afraid to generalize from examples (Stedall correctly calls this "induction," thereby taking the risk of confusing mathematicians unfamiliar with this use of the term). In the latter half of the book the idea of interpolation becomes central as he attempts the quadrature of the circle. The book concludes with what are essentially formulas for π, representing it both as an infinite product and as a continued fraction.

The Arithmetica Infinitorum was written when Newton was 13 and Leibniz was 10 years old, demonstrating once again that much of what we think of as "calculus" was discovered before there was any such subject. (The "Nova Methodus" of Wallis's subtitle is echoed — on purpose, perhaps? — in Leibniz's own "Nova Methodus" article, the first printed account of the new calculus, in 1684.) Stedall's translation gives us access once again to this fascinating book, and her introduction helps us understand its place in history. Not to be missed.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College; he thinks reprints of classic mathematics books are really cool.